Where: Here, pij is the probability that elements i and j will both be on when the system is in its training phase (positive phase), and qij is the probability that both elements i and j will be on during the production phase (negative phase).

Artificial Neural Networks/Boltzmann Machines

Where: Here, pij is the probability that elements i and j will both be on when the system is in its training phase (positive phase), and qij is the probability that both elements i and j will be on during the production phase (negative phase).

(Editor’s note: While RBMs are occasionally used, most practitioners in the machine-learning community have deprecated them in favor of generative adversarial networks or variational autoencoders.)

Each circle in the graph above represents a neuron-like unit called a node, and nodes are simply where calculations take place.

Each node is a locus of computation that processes input, and begins by making stochastic decisions about whether to transmit that input or not.

(Stochastic means “randomly determined”, and in this case, the coefficients that modify inputs are randomly initialized.) Each visible node takes a low-level feature from an item in the dataset to be learned.

(MNIST images have 784 pixels, so neural nets processing them must have 784 input nodes on the visible layer.) Now let’s follow that single pixel value, x, through the two-layer net.

The result of those two operations is fed into an activation function, which produces the node’s output, or the strength of the signal passing through it, given input x.

Each x is multiplied by a separate weight, the products are summed, added to a bias, and again the result is passed through an activation function to produce the node’s output.

That is, a single input x would have three weights here, making 12 weights altogether (4 input nodes x 3 hidden nodes).

The sum of those products is again added to a bias (which forces at least some activations to happen), and the result is passed through the activation algorithm producing one output for each hidden node.

But in this introduction to restricted Boltzmann machines, we’ll focus on how they learn to reconstruct data by themselves in an unsupervised fashion (unsupervised means without ground-truth labels in a test set), making several forward and backward passes between the visible layer and hidden layer no.

You can think of reconstruction error as the difference between the values of r and the input values, and that error is then backpropagated against the RBM’s weights, again and again, in an iterative learning process until an error minimum is reached.

As you can see, on its forward pass, an RBM uses inputs to make predictions about node activations, or the probability of output given a weighted x: p(a|x;

But on its backward pass, when activations are fed in and reconstructions, or guesses about the original data, are spit out, an RBM is attempting to estimate the probability of inputs x given activations a, which are weighted with the same coefficients as those used on the forward pass.

Reconstruction does something different from regression, which estimates a continous value based on many inputs, and different from classification, which makes guesses about which discrete label to apply to a given input example.

This is known as generative learning, which must be distinguished from the so-called discriminative learning performed by classification, which maps inputs to labels, effectively drawing lines between groups of data points.

KL-Divergence measures the non-overlapping, or diverging, areas under the two curves, and an RBM’s optimization algorithm attempts to minimize those areas so that the shared weights, when multiplied by activations of hidden layer one, produce a close approximation of the original input.

The hidden bias helps the RBM produce the activations on the forward pass (since biases impose a floor so that at least some nodes fire no matter how sparse the data), while the visible layer’s biases help the RBM learn the reconstructions on the backward pass.

This process of creating sequential sets of activations by grouping features and then grouping groups of features is the basis of a feature hierarchy, by which neural networks learn more complex and abstract representations of data.

It requires no labels to improve the weights of the network, which means you can train on unlabeled data, untouched by human hands, which is the vast majority of data in the world.

Because those weights already approximate the features of the data, they are well positioned to learn better when, in a second step, you try to classify images with the deep-belief network in a subsequent supervised learning stage.

Proper weight initialization can save you a lot of training time, because training a net is nothing more than adjusting the coefficients to transmit the best signals, which allow the net to classify accurately.

If a node passes the signal through, it is “activated.” optimizationAlgo refers to the manner by which a neural net minimizes error, or finds a locus of least error, as it adjusts its coefficients step by step.

LBFGS, an acronym whose letters each refer to the last names of its multiple inventors, is an optimization algorithm that makes use of second-order derivatives to calculate the slope of gradient along which coefficients are adjusted.

Regularization essentially punishes large coefficients, since large coefficients by definition mean the net has learned to pin its results to a few heavily weighted inputs.

The transformation is an additional algorithm that squashes the data after it passes through each layer in a way that makes gradients easier to compute (and gradients are necessary for a net to learn).